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Theorem 1stcclb 22623
Description: A property of points in a first-countable topology. (Contributed by Jeff Hankins, 22-Aug-2009.)
Hypothesis
Ref Expression
1stcclb.1 𝑋 = 𝐽
Assertion
Ref Expression
1stcclb ((𝐽 ∈ 1stω ∧ 𝐴𝑋) → ∃𝑥 ∈ 𝒫 𝐽(𝑥 ≼ ω ∧ ∀𝑦𝐽 (𝐴𝑦 → ∃𝑧𝑥 (𝐴𝑧𝑧𝑦))))
Distinct variable groups:   𝑥,𝑦,𝑧,𝐴   𝑥,𝐽,𝑦,𝑧   𝑥,𝑋,𝑦,𝑧

Proof of Theorem 1stcclb
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 1stcclb.1 . . . 4 𝑋 = 𝐽
21is1stc2 22621 . . 3 (𝐽 ∈ 1stω ↔ (𝐽 ∈ Top ∧ ∀𝑤𝑋𝑥 ∈ 𝒫 𝐽(𝑥 ≼ ω ∧ ∀𝑦𝐽 (𝑤𝑦 → ∃𝑧𝑥 (𝑤𝑧𝑧𝑦)))))
32simprbi 496 . 2 (𝐽 ∈ 1stω → ∀𝑤𝑋𝑥 ∈ 𝒫 𝐽(𝑥 ≼ ω ∧ ∀𝑦𝐽 (𝑤𝑦 → ∃𝑧𝑥 (𝑤𝑧𝑧𝑦))))
4 eleq1 2821 . . . . . . 7 (𝑤 = 𝐴 → (𝑤𝑦𝐴𝑦))
5 eleq1 2821 . . . . . . . . 9 (𝑤 = 𝐴 → (𝑤𝑧𝐴𝑧))
65anbi1d 629 . . . . . . . 8 (𝑤 = 𝐴 → ((𝑤𝑧𝑧𝑦) ↔ (𝐴𝑧𝑧𝑦)))
76rexbidv 3169 . . . . . . 7 (𝑤 = 𝐴 → (∃𝑧𝑥 (𝑤𝑧𝑧𝑦) ↔ ∃𝑧𝑥 (𝐴𝑧𝑧𝑦)))
84, 7imbi12d 344 . . . . . 6 (𝑤 = 𝐴 → ((𝑤𝑦 → ∃𝑧𝑥 (𝑤𝑧𝑧𝑦)) ↔ (𝐴𝑦 → ∃𝑧𝑥 (𝐴𝑧𝑧𝑦))))
98ralbidv 3168 . . . . 5 (𝑤 = 𝐴 → (∀𝑦𝐽 (𝑤𝑦 → ∃𝑧𝑥 (𝑤𝑧𝑧𝑦)) ↔ ∀𝑦𝐽 (𝐴𝑦 → ∃𝑧𝑥 (𝐴𝑧𝑧𝑦))))
109anbi2d 628 . . . 4 (𝑤 = 𝐴 → ((𝑥 ≼ ω ∧ ∀𝑦𝐽 (𝑤𝑦 → ∃𝑧𝑥 (𝑤𝑧𝑧𝑦))) ↔ (𝑥 ≼ ω ∧ ∀𝑦𝐽 (𝐴𝑦 → ∃𝑧𝑥 (𝐴𝑧𝑧𝑦)))))
1110rexbidv 3169 . . 3 (𝑤 = 𝐴 → (∃𝑥 ∈ 𝒫 𝐽(𝑥 ≼ ω ∧ ∀𝑦𝐽 (𝑤𝑦 → ∃𝑧𝑥 (𝑤𝑧𝑧𝑦))) ↔ ∃𝑥 ∈ 𝒫 𝐽(𝑥 ≼ ω ∧ ∀𝑦𝐽 (𝐴𝑦 → ∃𝑧𝑥 (𝐴𝑧𝑧𝑦)))))
1211rspcv 3559 . 2 (𝐴𝑋 → (∀𝑤𝑋𝑥 ∈ 𝒫 𝐽(𝑥 ≼ ω ∧ ∀𝑦𝐽 (𝑤𝑦 → ∃𝑧𝑥 (𝑤𝑧𝑧𝑦))) → ∃𝑥 ∈ 𝒫 𝐽(𝑥 ≼ ω ∧ ∀𝑦𝐽 (𝐴𝑦 → ∃𝑧𝑥 (𝐴𝑧𝑧𝑦)))))
133, 12mpan9 506 1 ((𝐽 ∈ 1stω ∧ 𝐴𝑋) → ∃𝑥 ∈ 𝒫 𝐽(𝑥 ≼ ω ∧ ∀𝑦𝐽 (𝐴𝑦 → ∃𝑧𝑥 (𝐴𝑧𝑧𝑦))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2101  wral 3059  wrex 3068  wss 3889  𝒫 cpw 4536   cuni 4841   class class class wbr 5077  ωcom 7732  cdom 8751  Topctop 22070  1stωc1stc 22616
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2103  ax-9 2111  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1540  df-ex 1778  df-sb 2063  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3060  df-rex 3069  df-rab 3224  df-v 3436  df-in 3896  df-ss 3906  df-pw 4538  df-uni 4842  df-1stc 22618
This theorem is referenced by:  1stcfb  22624  1stcrest  22632  lly1stc  22675  tx1stc  22829
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